nLab geometrically connected scheme

Definition

Definition

Let $k$ be a field and $X$ a $k$-scheme. We say that $X$ is geometrically connected if the scheme $X_{k'}$ is connected for every field extension $k'$ of $k$.

Properties

Proposition

Let $k^{\mathrm{sep}}$ be the separable closure of $k$. Then $X$ is geometrically connected if and only if the base change $X_{k^{\mathrm{sep}}}$ is connected.

References

• The Stacks Project[tag:0361]

category: algebraic geometry